arises in the linearization of the Vlasov-Poisson equation of plasma physics [1]. An application is to find the numerical abscissa of A; that is, the maximum eigenvalue of the compact operator B=(A+A')/2, where A' is the adjoint of A. The positivity of this quantity may give some indication of nonmodal transient growth of perturbations in equilibrium plasmas [1,2].

For a piecewise smooth kernel, the VOLT function can produce the original operator A directly:

Chebfun does not currently have a facility for finding the adjoint of a linear operator. For a Volterra operator, however, the adjoint can again be written in terms of integral operators using the kernel K. In this case, the symmetry property K(s,t)=K(t,s) simplifies things even further, and B is just the Fredholm variant of A. So we can use FRED:

B = 0.5*fred(K,d);

The following lines create a function to return the numerical abscissa given a value for a.

Note that while the eigenfunctions themselves vary smoothly with the parameter, the color scheme changes in the two plots! This indicates that the role of leading eigenfunction (blue) has passed from one to the other.